IDEAS home Printed from https://ideas.repec.org/a/spr/jotpro/v22y2009i3d10.1007_s10959-009-0233-7.html
   My bibliography  Save this article

On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains

Author

Listed:
  • James Allen Fill

    (The Johns Hopkins University)

Abstract

An (upward) skip-free Markov chain with the set of nonnegative integers as state space is a chain for which upward jumps may be only of unit size; there is no restriction on downward jumps. In a 1987 paper, Brown and Shao determined, for an irreducible continuous-time skip-free chain and any d, the passage time distribution from state 0 to state d. When the nonzero eigenvalues ν j of the generator on {0,…,d}, with d made absorbing, are all real, their result states that the passage time is distributed as the sum of d independent exponential random variables with rates ν j . We give another proof of their theorem. In the case of birth-and-death chains, our proof leads to an explicit representation of the passage time as a sum of independent exponential random variables. Diaconis and Miclo recently obtained the first such representation, but our construction is much simpler. We obtain similar (and new) results for a fastest strong stationary time T of an ergodic continuous-time skip-free chain with stochastically monotone time-reversal started in state 0, and we also obtain discrete-time analogs of all our results. In the paper’s final section we present extensions of our results to more general chains.

Suggested Citation

  • James Allen Fill, 2009. "On Hitting Times and Fastest Strong Stationary Times for Skip-Free and More General Chains," Journal of Theoretical Probability, Springer, vol. 22(3), pages 587-600, September.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-009-0233-7
    DOI: 10.1007/s10959-009-0233-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10959-009-0233-7
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10959-009-0233-7?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    2. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Michael C. H. Choi & Lu-Jing Huang, 2020. "On Hitting Time, Mixing Time and Geometric Interpretations of Metropolis–Hastings Reversiblizations," Journal of Theoretical Probability, Springer, vol. 33(2), pages 1144-1163, June.
    2. Laurent Miclo & Pierre Patie, 2021. "On interweaving relations," Post-Print hal-03159496, HAL.
    3. James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
    4. Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.
    5. L. Avena & A. Gaudillière, 2018. "Two Applications of Random Spanning Forests," Journal of Theoretical Probability, Springer, vol. 31(4), pages 1975-2004, December.
    6. Yong-Hua Mao & Feng Wang & Xian-Yuan Wu, 2015. "Large Deviation Behavior for the Longest Head Run in an IID Bernoulli Sequence," Journal of Theoretical Probability, Springer, vol. 28(1), pages 259-268, March.
    7. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.
    8. Persi Diaconis & Laurent Miclo, 2009. "On Times to Quasi-stationarity for Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 22(3), pages 558-586, September.
    9. Miclo, Laurent & Arnaudon, Marc & Coulibaly-Pasquier, Koléhè, 2024. "On Markov intertwining relations and primal conditioning," TSE Working Papers 24-1509, Toulouse School of Economics (TSE).
    10. Yu Gong & Yong-Hua Mao & Chi Zhang, 2012. "Hitting Time Distributions for Denumerable Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 950-980, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. James Allen Fill & Vince Lyzinski, 2014. "Hitting Times and Interlacing Eigenvalues: A Stochastic Approach Using Intertwinings," Journal of Theoretical Probability, Springer, vol. 27(3), pages 954-981, September.
    2. Yu Gong & Yong-Hua Mao & Chi Zhang, 2012. "Hitting Time Distributions for Denumerable Birth and Death Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 950-980, December.
    3. James Allen Fill & Vince Lyzinski, 2016. "Strong Stationary Duality for Diffusion Processes," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1298-1338, December.
    4. Miclo, Laurent & Arnaudon, Marc & Coulibaly-Pasquier, Koléhè, 2024. "On Markov intertwining relations and primal conditioning," TSE Working Papers 24-1509, Toulouse School of Economics (TSE).
    5. Erik A. Doorn, 2017. "An Orthogonal-Polynomial Approach to First-Hitting Times of Birth–Death Processes," Journal of Theoretical Probability, Springer, vol. 30(2), pages 594-607, June.
    6. James Allen Fill, 2009. "The Passage Time Distribution for a Birth-and-Death Chain: Strong Stationary Duality Gives a First Stochastic Proof," Journal of Theoretical Probability, Springer, vol. 22(3), pages 543-557, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jotpro:v:22:y:2009:i:3:d:10.1007_s10959-009-0233-7. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.